# d(y,x,n,a) is fractional differentiation of y with x in n order from a;


d(y_,x_,n_,0):=d(y,x,n);
d(e^(y_),x_,n_,-oo):=exp(y)*d(y,x)^n;
d(e^x_,x_,n_,-oo):=exp(x);

d(a_ and b_,x_,n_):=d(a,x,n) and d(b,x,n);
d(a_ and b_,x_):=d(a,x) and d(b,x);
d(a_=b_,x_,n_):=d(a,x,n)=d(b,x,n);
d(a_+b_,x_,n_):=d(a,x,n)+d(b,x,n);
d(a_+b_,x_):=d(a,x)+d(b,x);

d(-y_,x_,n_):= -d(y,x,n);

d(ds(y_,x_,n_),x_,m_) := ds(y,x,n+m);
d(d(y_,x_,n_),x_,m_) := d(y,x,n+m);
d(d(y_,x_),x_,n_) := d(y,x,n+1);
d(d(y_,x_,n_),x_) := d(y,x,n+1);
d(d(y_,x_),x_) := d(y,x,2);
d(integrate(y_,x_),x_, p_) := d(y,x,p-1);

d(psolution(c_,d_,y_,x_,n_),x_,n_):=d;
d(nsolve(y_),x_,n_):=0;

d(a_*x_^m_,x_,n_):= If(isfree(a,x), a*d(x^m,x,n), If(n>0,  
  if(isinteger(n), sum(binomial(n,k)*fallingfactorial(m,k)*d(a,x,n-k)*x^(m-k), k,0,n,1), 
  If(isinteger(m), sum(binomial(n,k)*fallingfactorial(m,k)*d(a,x,n-k)*x^(m-k), k,0,ceil(abs(m)),1) ))));
d(a_*(c_+x_)^m_,x_,n_):= If(isfree(a,c,x), y*d((c+x)^m,x,n), If(n>0,
  If(isinteger(n), sum(binomial(n,k)*fallingfactorial(m,k)*d(a,x,n-k)*(c+x)^(m-k), k,0,n,1), 
  If(isinteger(m), sum(binomial(n,k)*fallingfactorial(m,k)*d(a,x,n-k)*(c+x)^(m-k), k,0,ceil(abs(m)),1) ))));
d(a_*(c_+b_*x_)^m_,x_,n_):= If(isfree(a,b,c,x), a*d((c+b*x)^m,x,n), If(n>0,
  If(isinteger(n), b^m*sum(binomial(n,k)*fallingfactorial(m,k)*d(a,x,n-k)*(c/b+x)^(m-k), k,0,n,1), 
  If(isinteger(m), b^m*sum(binomial(n,k)*fallingfactorial(m,k)*d(a,x,n-k)*(c/b+x)^(m-k), k,0,ceil(abs(m)),1) ))));

d(x_*y_,x_,n_):=If(isfree(y,x),If(n>1,0), if(n>0,x*d(y,x,n)+n*d(y,x,n-1) ));
d(a_*x_,x_,n_):=If(isfree(a,x),If(n>1,0), if(n>= -1,x*d(a,x,n)+n*d(a,x,n-1) ));

d(a_^x_,x_,n_):= If(isfree(a,x), log(a)^n*a^x );
d((a_+x_^m_)^b_,x_,n_):= if(hasnot(a,b,x) and n>0 and n<=1, fallingfactorial(b,n)*(a+x^m)^(b-n)*m^n*x^(m*n-n));
d((a_+c_*x_^m_)^b_,x_,n_):= if(hasnot(a,b,c,x) and n>0 and n<=1, fallingfactorial(b,n)*(a+c*x^m)^(b-n)*c^n*m^n*x^(m*n-n));
d(c_^(a_*x_),x_,n_) := If(isfree(c,a,x),(a*log(c))^n*c^(a*x));
d(c_^(x_+b_),x_,n_) := If(isfree(c,b,x), log(c)^n*c^(x+b));
d(c_^(b_+a_*x_),x_,n_) := If(isfree(c,b,a,x), (a*log(c))^n*c^(a*x+b));

#d(e^f_*y_,x_,p_):=  If(p<=0 and p>= -1 and isfree(y*d(f,x)^p,x), e^f*y*d(f,x)^p, If(p>=0 and p<=1 and f<>x, d(f,x)^p*replace(d(e^x*y,x,p),x,f) ));
#d(exp(x_)*y_,x_,p_):=  If(p>=0 and p<=1, exp(x)*(p*d(y,x,p)+y));
#d(e^(a_+x_)*y_,x_,p_):= If(isfree(a,x), e^a*d(e^x*y,x,p));
#d(exp(a_+x_)*y_,x_,p_):= If(isfree(a,x), exp(a)*d(e^x*y,x,p));


d(e^(x_^m_)*x_,x_,n_):=If(n>= -1 and n<0 and m*n==(n-1), exp(x^m)*m^n);
d(e^(-x_^m_)*x_,x_,n_):=If(n>= -1 and n<0 and m*n==(n-1), exp(-x^m)*(-m)^n);

d(e^(a_*x_),x_,n_) := If(isfree(a,x), a^n*e^(a*x));
d(e^(x_+b_),x_,n_) := If(isfree(b,x), e^(x+b));
d(e^(b_+a_*x_),x_,n_) := If(isfree(b,a,x), a^n*e^(a*x+b));

#d(x_^m_,x_,n_) := If(isfree(m,x), If(m<0 and isinteger(m) and (m>=n or n<m+1), -(-1)^m/(-1-m)!/(m-n)!*x^(m-n)*log(x), fallingfactorial(m,n)*x^(m-n) ));
#d((c_+x_)^m_,x_,n_) := If(isfree(c,m,x), If(m== -1 and n<0, log(c+x)*(c+x)^(-1-n)/(-1-n)!, fallingfactorial(m,n)*(c+x)^(m-n) ));
#d((a_*x_)^m_,x_,n_) := If(isfree(a,m,x), If(m== -1 and n<0, log(x)*x^(-1-n)/(-1-n)!*a^m, fallingfactorial(m,n)*x^(m-n)*a^m ));
#d((c_+a_*x_)^m_,x_,n_) := If(isfree(c,a,m,x), If(m== -1 and n<0, log(c+a*x)*(c+a*x)^(-1-n)/(-1-n)!*a^n, fallingfactorial(m,n)*(c+a*x)^(m-n)*a^n ));

#d(x_^m_,x_,n_) := If(isfree(m,x), If(m== -1 and n<=0, ln(1+n,x), If(m<0 and (m>=n or n<m+1), -(-1)^m/(-1-m)!/(m-n)!*x^(m-n)*log(x), fallingfactorial(m,n)*x^(m-n) )));
d(x_^m_,x_,n_) := If(isfree(m,x), If(m== -1 and n<0, ln(-n,x), If(m<0 and m==n, (-1)^(1+n)/(-1-n)!*log(x), fallingfactorial(m,n)*x^(m-n) )));
d((c_+x_)^m_,x_,n_) := If(isfree(c,m,x), If(m== -1 and n<0, ln(-n,c+x), If(m<0 and m==n, -(-1)^n*log(c+x)/(-n)!, fallingfactorial(m,n)*(c+x)^(m-n) )));
d((a_*x_)^m_,x_,n_) := If(isfree(a,m,x), If(m== -1 and n<0, a^m*ln(-n,x), If(m<0 and m==n, -(-1)^n*log(x)/(-n)!,fallingfactorial(m,n)*x^(m-n)*a^m )));
d((c_+a_*x_)^m_,x_,n_) := If(isfree(c,a,m,x), If(m== -1 and n<0, ln(-n,c+a*x), If(m<0 and m==n, -(-1)^n*log(c+a*x)/(-n)!,fallingfactorial(m,n)*(c+a*x)^(m-n)*a^n )));

d(a_*sinh(x_),x_,p_):= d(a*e^x/2-a*e^(-x)/2,x,p);
d(a_*cosh(x_),x_,p_):= d(a*e^x/2+a*e^(-x)/2,x,p);
d(sinh(x_)*f_^(x_),x_,n_):= If(isfree(f,x), e^((log(f)+1)*x)*(log(f)+1)^n/2-d(e^((log(f)-1)*x),x,n)/2 );
d(cosh(x_)*f_^(x_),x_,n_):= If(isfree(f,x), e^((log(f)+1)*x)*(log(f)+1)^n/2+d(e^((log(f)-1)*x),x,n)/2 );
d(sinh(x_)*f_^(-x_),x_,n_):= If(isfree(f,x), d(e^((1-log(f))*x),x,n)/2-e^((-log(f)-1)*x)*(-log(f)-1)^n/2 );
d(cosh(x_)*f_^(-x_),x_,n_):= If(isfree(f,x), d(e^((1-log(f))*x),x,n)/2+e^((-log(f)-1)*x)*(-log(f)-1)^n/2 );

d(a_*b_*c_,x_,p_):=If(isfree(a,x), a*d(b*c,x,p), If(isfree(b,x), b*d(a*c,x,p) ));

#d(sin(x_)*e^(x_),x_,n_):=sin(x+n*pi/4)*e^(x)*2^(n/2);
#d(cos(x_)*e^(x_),x_,n_):=cos(x+n*pi/4)*e^(x)*2^(n/2);
d(sin(x_)*exp(x_),x_,n_):=sin(x+n*pi/4)*exp(x)*2^(n/2);
d(cos(x_)*exp(x_),x_,n_):=cos(x+n*pi/4)*exp(x)*2^(n/2);
d(sin(a_*x_)*e^(x_),x_,n_):=If(isfree(a,x), sin(a*x+n*atan(a))*e^(x)*(a^2+1)^(n/2));
d(cos(a_*x_)*e^(x_),x_,n_):=If(isfree(a,x), cos(a*x+n*atan(a))*e^(x)*(a^2+1)^(n/2));
d(sin(x_+b_)*e^(x_),x_,n_):=If(isfree(b,x), sin(x+n*pi/4+b)*e^(x)*2^(n/2));
d(cos(x_+b_)*e^(x_),x_,n_):=If(isfree(b,x), cos(x+n*pi/4+b)*e^(x)*2^(n/2));
d(sin(a_*x_)*e^(b_*x_),x_,n_):=If(isfree(a,b,x), sin(a*x+n*atan(a/b))*e^(b*x)*(a^2+b^2)^(n/2));
d(cos(a_*x_)*e^(b_*x_),x_,n_):=If(isfree(a,b,x), cos(a*x+n*atan(a/b))*e^(b*x)*(a^2+b^2)^(n/2));
d(sin(x_)*e^(b_*x_),x_,n_):=If(isfree(b,x), sgn(n)*sin(x+n*atan(1/b))*e^(b*x)*(1+b^2)^(n/2));
d(cos(x_)*e^(b_*x_),x_,n_):=If(isfree(b,x), sgn(n)*cos(x+n*atan(1/b))*e^(b*x)*(1+b^2)^(n/2));
d(sin(x_+c_)*e^(b_*x_),x_,n_):=If(isfree(c,b,x), sin(c+x+n*atan(1/b))*e^(b*x)*(1+b^2)^(n/2));
d(cos(x_+c_)*e^(b_*x_),x_,n_):=If(isfree(c,b,x), cos(c+x+n*atan(1/b))*e^(b*x)*(1+b^2)^(n/2));
d(e^(x_)*sin(a_*x_+b_),x_,n_):=If(isfree(a,b,x), sin(a*x+n*atan(a)+b)*e^(x)*(a^2+1)^(n/2) );
d(e^(x_)*cos(a_*x_+b_),x_,n_):=If(isfree(a,b,x), cos(a*x+n*atan(a)+b)*e^(x)*(a^2+1)^(n/2) );
d(e^(c_*x_)*sin(a_*x_+b_),x_,n_):=If(isfree(a,b,c,x), sin(a*x+b+n*atan(a/c))*e^(x)*(a^2+c^2)^(n/2) );
d(e^(c_*x_)*cos(a_*x_+b_),x_,n_):=If(isfree(a,b,c,x), cos(a*x+b+n*atan(a/c))*e^(x)*(a^2+c^2)^(n/2) );

d((x_+b_)^k_*(x_+d_)^m_,x_,n_):= If(isfree(b,k,d,m,x), sum(fallingfactorial(k,n-r)*fallingfactorial(m,r)*binomial(n,r)*(x+b)^(k-n+r)*(x+d)^(m-r),r,0,ceil(abs(m)),1));
d((a_*x_+b_)^k_*(c_*x_+d_)^m_,x_,n_):= If(isfree(a,b,k,c,d,m,x), sum(binomial(n,r)*a^(r-n)*c^r*fallingfactorial(k,n-r)*fallingfactorial(m,r)*(a*x+b)^(k-n+r)*(c*x+d)^(m-r),r,0,ceil(abs(m)),1));
d((x_+b_)^k_*(c_*x_+d_)^m_,x_,n_):= If(isfree(b,k,c,d,m,x), sum(binomial(n,r)*c^r*fallingfactorial(k,n-r)*fallingfactorial(m,r)*(x+b)^(k-n+r)*(c*x+d)^(m-r),r,0,ceil(abs(m)),1));
d(x_^k_*(x_+d_)^m_,x_,n_):= If(isfree(k,d,m,x), sum(binomial(n,r)*fallingfactorial(k,n-r)*fallingfactorial(m,r)*x^(k-n+r)*(x+d)^(m-r),r,0,ceil(abs(m)),1));
d(x_^k_*(c_*x_+d_)^m_,x_,n_):= If(isfree(k,c,d,m,x), sum(binomial(n,r)*c^r*fallingfactorial(k,n-r)*fallingfactorial(m,r)*x^(k-n+r)*(c*x+d)^(m-r),r,0,ceil(abs(m)),1));

#d((c_+x_^2)^k_,x_,n_) := If(isfree(c,x), d((sqrt(-c)+x)^k*(-sqrt(-c)+x)^k,x,n));
#d((c_-x_^2)^k_,x_,n_) := If(isfree(c,x), d((sqrt(c)+x)^k*(sqrt(c)-x)^k,x,n));
d((k_+x_^2)^(-0.5),x_,n_) := If(isfree(k,x), asinh(n+1,x/sqrt(k)) );
d(1/(k_+x_^2),x_,n_) := If(isfree(k,x), If(k<0, -atanh(1+n,x/sqrt(-k))/sqrt(-k), atan(1+n,x/sqrt(k))/sqrt(k)  ));
d(1/(k_-x_^2),x_,n_) := If(isfree(k,x), If(k<0, -atan(1+n,x/sqrt(-k))/sqrt(-k), atanh(1+n,x/sqrt(k))/sqrt(k) ));

d(log(x_)*x_,x_,n_) := If(n>1, (-1)^n*(n-2)!*x^(1-n),  (psi(2)-psi(2-n)+log(x))/Gamma(2-n)*x^(1-n));
#d(log(x_)*x_^m_,x_,n_) := If(n>= -1 and n<=0, If(n==m, -(-1)^n*(m+1)^(-2)*log(x)^(1-m), If(m== -1, 1/2*d(log(x)^2,x,n+1),  -(m+1)^(-2)*Gamma(1-n, (-m+n)*log(x)) )), -(m+1)^(-2)*d(Gamma(2, -(m+1)*log(x)),x,1+n));
d(log(x_)/x_,x_,n_):=If(n<0,1/2*d(log(x)^2,x,n+1));
#d(x_^m_*log(x_),x_,n_) :=  If(m== -1, 1/2*d(log(x)^2,x,n+1), If(n>= -1 and n<=0, If(n==m, -(-1)^n*(m+1)^(-2)*log(x)^(1-m),  -(m+1)^(-2)*Gamma(1-n, (-m+n)*log(x)) )), -(m+1)^(-2)*d(Gamma(2, -(m+1)*log(x)),x,1+n));
d(x_^m_*log(x_),x_,n_) := If(n>= -1 and n<=0, If(n==m,log(x)^(1-m), If(m== -1, 1/2*d(log(x)^2,x,n+1),  -(m+1)^(-2)*Gamma(1-n, (-m+n)*log(x)) )),
  If(isinteger(n-m) and n>m, -(-1)^(n-m)*(n-m-1)!*Gamma(1+m)*x^(m-n), 
  If(m> -1, fallingfactorial(m,n)*(psi(1+m)-psi(1+m-n)+log(x))*x^(m-n), 
  If(m< -1, fallingfactorial(m,n)*(psi(abs(m))-psi(abs(m-n))+log(x))*x^(m-n) ))));
#d(1/log(y_),x_,p_):=li(p+1,y);
d(log(x_)^m_,x_,p_):=  If(m== -1, li(p+1,x), If(abs(m)==abs(p), (-1)^(-m)*d(Gamma(m+1, -log(x)),x,1+p), If(p> -1 and p<=0,  (-1)^(-m)*Gamma(m-p, p*log(x)) )));
d(log(x_+c_)^m_,x_,p_):=  If(isfree(c,x),If(m= -1, li(p+1,x+c), If(p> -1 and p<=0, (-1)^(-m)*Gamma(m-p, p*log(x+c)) )));
d(log(x_)^m_*x_^n_,x_,p_):= if(n== -1 and m== -1, ln(1+p,log(x)), If(p>= -1 and p<=0, If(n== -1, 1/(m+1)*d(log(x)^(m+1),x,p+1), 
  If(n==p, (-1)^(m-p)*(-n-1)^(-m)/(n+1)*log(x)^(m-p), (-n-1)^(-m)/(n+1)*Gamma(m-p, (-n+p)*log(x)) )), (-n-1)^(-m)/(n+1)*d(Gamma(m+1, (-n-1)*log(x)),x,1+p) ));
#d(x_^n_*log(x_)^m_,x_,p_):= If(p>= -1 and p<=0, If(n==-1, 1/(m+1)*d(log(x)^(m+1),x,p+1), 
  If(n==p, (-1)^(-m)*(-n-1)^(-m)/(n+1)*log(x)^(m-p), 1/(1+n)*d(Gamma(m+1, (-n-1)*log(x)),x,p+1) )));

d(e^x_*log(x_),x_,n_):= If(n>1, exp(x)*log(x)+exp(x)*sum((-1)^(1+k)*binomial(n,k)*Gamma(k)*x^(-k), k,1,n,1) );
d(e^(-x_)*log(x_),x_,n_):= if(n>1, (-1)^n*exp(-x)*(log(x)-sum(binomial(n,k)*Gamma(k)*x^(-k), k,1,n,1)) ); 

d(e^x_*x_^m_,x_,n_):= If(n== -m, exp(x)*(x^m-m!),If(n==m, exp(x)*(x^m+m!),If(n>0, if(isinteger(n), exp(x)*sum(binomial(n,k)*fallingfactorial(m,k)*x^(m-k), k,0,n,1), 
  If(isinteger(m), exp(x)*sum(binomial(n,k)*fallingfactorial(m,k)*x^(m-k), k,0,abs(m),1) )))));
#d(e^(a_*x_)*x_^m_,x_,n_):= If(isfree(m,a,x), If(n>0 and isinteger(n), exp(a*x)*sum(a^(n-k)*binomial(n,k)*fallingfactorial(m,k)*x^(m-k), k,0,n,1), 
  If(m>0 and n> -1 and isinteger(m), exp(a*x)*sum(a^(n-k)*binomial(n,k)*fallingfactorial(m,k)*x^(m-k), k,0,abs(m),1), 
  If(m== -1, Ei(1+n,a*x), (-a)^(-m)*Gamma(n+1,m+1, -a*x)  ))));
d(e^(a_*x_)*x_^m_,x_,n_):= If(isfree(m,a,x), If(n>0, if(isinteger(n), exp(a*x)*sum(a^(n-k)*binomial(n,k)*fallingfactorial(m,k)*x^(m-k), k,0,n,1), 
  If(isinteger(m), exp(a*x)*sum(a^(n-k)*binomial(n,k)*fallingfactorial(m,k)*x^(m-k), k,0,abs(m),1), 
  If(m== -1, Ei(1+n,a*x)  )))));
d(e^(c_+a_*x_)*x_^m_,x_,n_):= If(isfree(m,c,a,x), If(n>0, if(isinteger(n), exp(c+a*x)*sum(a^(n-k)*binomial(n,k)*fallingfactorial(m,k)*x^(m-k), k,0,n,1), 
  If(isinteger(m), exp(c+a*x)*sum(a^(n-k)*binomial(n,k)*fallingfactorial(m,k)*x^(m-k), k,0,abs(m),1),
  If(m== -1, exp(c)*Ei(1+n,a*x)  )))));
d(e^x_*(c_+x_)^m_,x_,n_):= If(isfree(c,m,x) and n>0, if(isinteger(n), exp(x)*sum(binomial(n,k)*fallingfactorial(m,k)*(c+x)^(m-k), k,0,n,1), 
  If(isinteger(m), exp(x)*sum(binomial(n,k)*fallingfactorial(m,k)*(c+x)^(m-k), k,0,abs(m),1) )));
d(e^x_*(c_+b_*x_)^m_,x_,n_):= If(isfree(b,c,m,x) and n>0, if(isinteger(n), b^m*exp(x)*sum(binomial(n,k)*fallingfactorial(m,k)*(c/b+x)^(m-k), k,0,n,1), 
  If(isinteger(m), b^m*exp(x)*sum(binomial(n,k)*fallingfactorial(m,k)*(c/b+x)^(m-k), k,0,abs(m),1) )));
d(e^x_*x_,x_,p_) := e^x*x+p*e^x;
d(e^x_/x_,x_,p_) := Ei(1+p,x);
d(e^x_/(c_+x_),x_,p_) := if(hasnot(c,x), e^(-c)*Ei(1+p,c+x));
d(e^x_/(c_+b_*x_),x_,p_) := if(hasnot(b,c,x), e^(-c)/b*Ei(1+p,c+b*x));
#d(exp(x_)*x_^a_,x_,a2_)  := If(a2== -a, exp(x)*(x^a-a!),If(a2==a, exp(x)*(x^a+a!) ));
d(exp(-x_)*x_^a_,x_,a2_) := If(a2== -a, -i*exp(-x)*(x^a-a!),If(a2==a, i*exp(-x)*(x^a+a!) ));

d(sin(x_)*x_^m_,x_,p_):= If(p>-1 and p<0, -(-i)^(-m)* d(Gamma(1 + m, (-i)* x),x,1+p)/2 -  d(Gamma(1 + m, i* x),x,p+1)*i^(-m)/2,  if(p>0 and isinteger(m),sum(binomial(p,k)*fallingfactorial(m,k)*x^(m-k)*sin(x+(p-k)*pi/2), k,0,abs(m),1) ));
d(cos(x_)*x_^m_,x_,p_):= If(p>-1 and p<0, -(-i)^(-m)* d(Gamma(1 + m, (-i)* x),x,1+p)/2 +  d(Gamma(1 + m, i* x),x,p+1)*i^(-m)/2,  if(p>0 and isinteger(m),sum(binomial(p,k)*fallingfactorial(m,k)*x^(m-k)*cos(x+(p-k)*pi/2), k,0,abs(m),1) ));
d(sinh(x_)*x_^m_,x_,p_):= If(p>-1 and p<0, (-1)^(-m)* d(Gamma(1 + m, -x),x,1+p)/2 +  d(Gamma(1 + m,  x),x,p+1)/2,  if(p>0 and isinteger(m),sum(binomial(p,k)*fallingfactorial(m,k)*x^(m-k)*sinh(x+(p-k)*pi*i/2), k,0,abs(m),1) ));
#d(cosh(x_)*x_^m_,x_,p_):= If(m<0, (-1)^(-m)* d(Gamma(1 + m, -x),x,1+p)/2 -  d(Gamma(1 + m,  x),x,p+1)/2, sum(binomial(p,k)*fallingfactorial(m,k)*x^(m-k)*cosh(x+(p-k)*pi*i/2), k,0,m,1));
d(cosh(x_)*x_^m_,x_,p_):= If(p>-1 and p<0,  (-1)^(-m)*d(Gamma(1 + m, -x),x,1+p)/2 -  d(Gamma(1 + m,  x),x,p+1)/2, if(p>0 and isinteger(m), sum(binomial(p,k)*fallingfactorial(m,k)*x^(m-k)*cosh(x+(p-k)*pi*i/2), k,0,abs(m),1) ));
#d(cosh(x_)*x_^m_,x_,p_):= if(p>0, if(isinteger(p), sum(binomial(p,k)*fallingfactorial(m,k)*x^(m-k)*cosh(x+(p-k)*pi*i/2), k,0,p,1),
  If(isinteger(m),sum(binomial(p,k)*fallingfactorial(m,k)*x^(m-k)*cosh(x+(p-k)*pi*i/2), k,0,abs(m),1) )));

d(e^(x_^m_),x_,p_):= If(p>= -1 and p<=0, (-1)^(p/m-p)*m^p*Gamma(1+p-p/m, -x^m), If(p>=0 and p<=1, (m*x^(m-1))^p*e^(x^m) ));
d(e^(a_*x_^m_),x_,p_):= If(p>= -1 and p<=0, (-a)^(p/m-p)*(a*m)^p*Gamma(1+p-p/m, -a*x^m), If(p>=0 and p<=1, (a*m*x^(m-1))^p*e^(a*x^m) ));
d(e^(x_^m_)*x_^n_,x_,p_):= If(p>= -1 and p<0, If(n==p*(1-m), exp(x^m)*m^p, -(-1)^((n-p)/(p*m))*m^p*Gamma((n-p)/(-p*m),-x^m) ));
#d(e^(x_^m_)*x_^n_,x_,p_):= If(p>= -1 and p<0 and n==p*(1-m), exp(x^m)*m^p, -(-1)^((-1-n)/m)/m*Gamma(p+1,(n+1)/m,-x^m) );
#d(e^(a_*x_^m_)*x_^n_,x_,p_):= If(isfree(a,x), If(p>= -1 and p<0 and n==p*(1-m), exp(a*x^m)*(a*m)^p, -(-a)^((-1-n)/m)/m*Gamma(p+1,(1+n)/m, -a*x^m) ));
d(e^(a_*x_^m_)*x_^n_,x_,p_):= If(isfree(a,x), If(p>= -1 and p<0, if(n==p*(1-m), exp(a*x^m)*(a*m)^p, -(-a)^((n-p)/(p*m))*m^p*Gamma((n-p)/(-p*m),-a*x^m) )));

#d(e^x_,x_,p_):=if(p>0 and p<1,exp(p,x));

d(mittag(a_,a_,c_*x_^a2_),x_,a_) := If(isfree(c,x) and a2== -a, c*x^(-2a)*mittag(a,a,c*x^a2) );
d(mittag(a_,a_,x_^a2_),x_,a_) := If( a2== -a, x^(-2a)*mittag(a,a,x^a2) );
d(mittag(a_,b_,x_^a_),x_,a_) :=  1/x^a/Gamma(b-a)+mittag(a,b,x^a);
d(mittag(a_,b_,c_*x_^a_),x_,a_) := If(isfree(c,x), 1/x^a/Gamma(b-a)+c*mittag(a,b,c*x^a) );
d(mittag(a_,b_,x_^a_)*x_^d_,x_,a2_) := If(d==b-1, If(a2==a,x^(b-a-1)/Gamma(b-a)+x^d*mittag(a,b,x^a),If(a2== -a,x^(a+b-1)*mittag(a,a+b,x^a),x^(d-a)*mittag(a,b-a,x^a) )));
d(mittag(a_,b_,c_*x_^a_)*x_^d_,x_,a2_) := If(isfree(c,x) and d==b-1, If(a2==a,x^(b-a-1)/Gamma(b-a)+c*x^d*mittag(a,b,c*x^a), 
	If(a2== -a,x^(a+b-1)*mittag(a,a+b,c*x^a),c*x^(d-a)*mittag(a,b-a,c*x^a) )));

d(c_*mittag(p_,f_),x_, p2_):=  If(p2<0 and p2> -1 and isfree(d(f,x,p)/c,x),  mittag(p,f)/d(f,x,p)*c*p!,if(p2==p,  c*d(f,x,p)*mittag(p,f)/(p!)+d(c,x,p2)*mittag(p,f) ));
d(mittag(p_,f_)*y_,x_, p2_):=  If(p2<0 and p2> -1 and isfree(d(f,x,p)/y,x),  mittag(p,f)/d(f,x,p)*y*p!, if(p2==p,  y*d(f,x,p)*mittag(p,f)/(p!)+d(y,x,p2)*mittag(p,f) ));
d(mittag(p_,f_),x_,p_):= if(p>0 and p<1, d(f,x,p)*mittag(p,f)/(abs(p))!);
d(mittag(a_,c_*x_^a_),x_,a2_) := If(isfree(c,x),if(a2==a or a2== -a, c^sgn(a*a2)*mittag(a,c*x^a) ));
d(mittag(a_,x_^a_),x_,a2_) := If(abs(a2)==abs(a), mittag(a,x^a));
#d(mittag(a_,c_*x_^a_*y_),x_,a2_) := If(isfree(c,y,x) and a==a2, c*mittag(a,c*x^a)*y,  c^sgn(a2)*mittag(a,c*x^a)*y^sgn(a2) );
#d(c_*mittag(p_,f_),x_, p2_):=  If(p2<0 and p2> -1 and isfree(d(f,x,p)/c,x),  mittag(p,f)/d(f,x,p)*c*p!, if(p2==p,  c*d(f,x,p)*mittag(p,f)/(p!)+d(c,x,p2)*mittag(p,f) ));
d(mittag(a_,x_^a_)*x_^a_,x_,a2_)  := If(a2== -a, E(a,x^a)*(x^a-a!),If(a2==a, E(a,x^a)*(x^a+a!) ));
d(mittag(a_,-x_^a_)*x_^a_,x_,a2_) := If(a2== -a, -E(a,-x^a)*(x^a+a!),If(a2==a, -E(a,-x^a)*(x^a-a!) ));
d(mittag(a_,x_^a_)*x_,x_,a2_)  := If(a2== -a, E(a,x^a)*(x-x^a/a!+1),If(a2==a, E(a,x^a)*(x+x^a/a!-1) ));
d(mittag(a_,-x_^a_)*x_,x_,a2_) := If(a2== -a, -E(a,-x^a)*(x+x^a/a!+1),If(a2==a, -E(a,-x^a)*(x-x^a/a!-1) ));

#d(Gamma(a_,m_,x_), x_,p_) := Gamma(a+p,m,x);
#d(Gamma(a_,x_,0),x_,p_):=Gamma(a+p,x,0);
#d(Gamma(a_,x_), x_,p_) := -d(exp(-x)*(x)^(a-1),x,p-1);
#d(Gamma(a_,x_), x_,n_) :=  if(n>0 and (n)<1,(-1)^n* x^(-n)* sum(binomial(n, k) *Gamma(a - k + n, x)* risingfactorial(a,k),k,0,n));
#d(Gamma(a_,x_), x_,n_) := if(abs(n)<1, Gamma(n,a,x));
#d(Gamma(x_), x_,p_) := if(p>0 and p<1, Gamma(x)*sum(binomial(p-1,k)*psi(x)^(p-1)*psi(k,x)^(k),k,0,p));
#d(Gamma(x_), x_,p_) := Gamma(p,x,0);
d(Gamma(n_,a_*log(x_)),x_,p_) := If(a+p<>0 and ((p> -1 and p<=0)  or n==p), Gamma(n-p, (a+p)*log(x)) );
d(Gamma(n_,a_*log(c_+x_)),x_,p_) := If(a+p<>0 and ((p> -1 and p<=0)  or n==p), Gamma(n-p, (a+p)*log(c+x)) );
d(Gamma(n_,log(x_)),x_,p_) := If(p> -1 and p<=0, Gamma(n-p, (1+p)*log(x)), If(p>=0 and p<=1,  (-1)^(n-p)*x^(-1-p)*log(x)^(n-p) ));
#d(Gamma(n_,a_*log(x_)),x_,p_) := If(p>= -1 and p<=0, d(log(x)^n/n+x^(-p)*Gamma(n, a*log(x)),x,1+p), If(p>=0 and p<=1,  fallingfactorial(n,p)*(-a^n)*x^(-p-a)*log(x)^(n-p)/n ));
#d(Gamma(n_,log(x_)),x_,p_) := If(p>= -1 and p<=0, d(log(x)^n/n+x^(-p)*Gamma(n, log(x)),x,1+p), If(p>=0 and p<=1,  (-1)^p*x^(-2p)*log(x)^(n-p) ));
d(Gamma(n_,x_^m_),x_,p_):= If(p>= -1 and p<=0, (-m)^p*Gamma(n+p-p/m,x^m) );
d(Gamma(n_,a_*x_^m_),x_,p_):=  If(p>= -1 and p<=0, a^(-1/m)*(-m)^p*Gamma(n+p-p/m,a*x^m), If(p>0 and p<=1 and isfree(a,x) and n+p-p/m==1, (a)^(p/m-p)*(-a*m)^p*e^(-a*x^m) ));
d(Gamma(x_), x_,p_) := infints(exp((-t))*t^(-1+x)*log(t)^p,t);
d(factorial(x_), x_,p_) := infints(exp((-t))*t^x*log(x)^p,t);

d(zeta(x_),x_,p_):= If(p>0,(-1)^p*sum(log(k)^p/k^x,k,2,oo));
d(zeta(m_,x_),x_,p_):= (-1)^p*risingfactorial(m,p)*zeta(m+p,x);
d(li(x_),x_,p_):= If(abs(p)<= 1,(-1)^p*Gamma(-p, (p-1)*log(x)) );
d(En(x_),x_,p_):= (-1)^p*En(1-p, -x);
d(En(m_, x_), x_,p_) := (-1)^p*En(m-p,-x);
d(erf(x_),x_,p_):= If(abs(p)<= 1, -(-2)^p*Gamma((p+1)/2,x^2)/sqrt(pi) );
d(erfc(x_),x_,p_):= If(abs(p)<= 1, (-2)^p*Gamma((p+1)/2,x^2)/sqrt(pi) );
d(erfi(x_),x_,p_):= If(abs(p)<= 1, (-2)^p*Gamma((p+1)/2,-x^2)/sqrt(pi) );
d(loggamma(x_),x_,p_) := psi(p-1,x);
d(sin(x_),x_,n_) := If(iseven(n/2), sin(x),sin(x+pi/2*n));
d(cos(x_),x_,n_) := If(iseven(n/2), cos(x),cos(x+pi/2*n));
d(sinh(x_),x_,n_) := If(iseven(n), sinh(x), If(isodd(n), cosh(x), sinh(x+n*pi*i/2) ));
d(cosh(x_),x_,n_) := If(iseven(n), cosh(x), If(isodd(n), sinh(x), cosh(x+n*pi*i/2) ));
d(abs(y_),x_,n_) := d(y,x,n)/sgn(y);
d(abs(x_),x_,n_) :=  If(n>1,0, If(n<0, 1/sgn(x)/(-n)!*x^(1-n), 1/sgn(x)*(1-n)!*x^(1-n) ));
d(sgn(y_),x_,n_) := If(n>0,0, If(n<0, 1/Gamma(1-n)*sgn(y)*x^(-n) ));
d(csgn(y_),x_,n_) := If(n>0,0, If(n<0, csgn(y)*x^(-n)/(-n)! ));
#d(delta(y_),x_,n_):=If(n>0,0, If(n<0, theta(y)*x^(-n)/(-n)! ));
d(theta(y_),x_,n_):= delta(n-1,y);
d(log(xx_),x,n_):= If(abs(n)<1 and has(xx,x),ln(n,xx));
d(log(x_),x_,n_):= If(n>0, -(-1)^n*(n-1)!/x^n, If(abs(n)<1,expand((log(x)+psi(1)-psi(1-n))/Gamma(1-n)/x^n) ));
d(log(c_+x_),x_,n_):= if(hasnot(c,x),If(n>0, -(-1)^n*(n-1)!/(c+x)^n, If(abs(n)<1,expand((log(c+x)+psi(1)-psi(1-n))/Gamma(1-n)/(c+x)^n) )));
#d(log(x_),x_,n_):= If(n>0, -(-1)^n*(n-1)!/x^n, If(n>= -1 and n<0, -Gamma(1-n, n*log(x)) ));
#d(log(x_),x_,n_):= If(n>0 and isinteger(n), (-1)^(n+1)*(n-1)!/x^n,  If(n>= -1 and n<0, -Gamma(1-n, n*log(x)) ));
#d(log(a_*x_+b_),x_,n_):= If(isfree(a,b,x),If(n>0 and isinteger(n), -(-1)^n*(n-1)!/(x+b/a)^n,If(abs(n)<=1,(log(x+b/a)+psi(1)-psi(1-n))/Gamma(1-n)/(x+b/a)^n ) ));
d(gauss(x_),x_,p_) := If(p>=0 and p<=1,(-x)^p*gauss(x));

d(asin(x_),x_,n_):=if(n>0 and isinteger(n), sum(binomial(n-1,n-1-2k)*(2k-1)!!*(2n-2k-3)!!*x^(n-2k-1)*(1-x^2)^(k-n+1/2),k,0,floor(n/2),1) );
d(acos(x_),x_,n_):=if(n>0 and isinteger(n), -sum((-1)^k*binomial(n-1,k)*(2k-1)!!*(2n-2k-3)!!/(1+x)^k*(1-x)^(k-n+1),k,0,n-1,1)/2^(n-1)/sqrt(1-x^2) );
d(atan(x_),x_,n_):= if(n>0, 1/2 i *(-1)^n *((-i + x)^(-n) - (i + x)^(-n))* (-1 + n)!, if(n<0 and isinteger(n),(psi(1)-psi(-n)+log(x-i))*(x-i)^(-n)/(-n)!/2-(psi(1)-psi(-n)+log(i+x))*(i+x)^(-n)/(-n)!/2 ));
d(acot(x_),x_,n_):= if(n>0, 1/2 i *(-1)^n *(-(-i + x)^(-n) + (i + x)^(-n))* (-1 + n)!,if(n<0 and isinteger(n),(psi(1)-psi(-n)+log(x+i))*(x+i)^(-n)/(-n)!/2-(psi(1)-psi(-n)+log(x-i))*(x-i)^(-n)/(-n)!/2 ));

d(asinh(x_),x_,n_):=if(n>0 and isinteger(n), (-1)^(n-1)*sum((-1)^k*binomial(n-1,n-1-2k)*(2k-1)!!*(2n-2k-3)!!*x^(n-2k-1)*(x^2-1)^(k-n+1/2),k,0,floor(n/2),1) );
d(acosh(x_),x_,n_):=if(n>0 and isinteger(n), (-1/2)^(n-1)*sum(binomial(n-1,k)*(2k-1)!!*(2n-2k-3)!!/(1+x)^(-k-1/2)*(x-1)^(k-n+1/2),k,0,n-1,1) );
#d(atanh(x_),x_,n_):= if(n>0 and isinteger(n),(psi(1)-psi(-n)+log(x+1))*(x+1)^(-n)/(-n)!/2+(psi(1)-psi(-n)+log(1-x))*(1-x)^(-n)/(-n)!/2);
d(atanh(x_),x_,n_):= if(n>0 and isinteger(n),(-1)^n*(n-1)!/(x^2+1)^n*sum(binomial(n,n+1-2k)*x^(n+1-2k),k,1,ceil(n/2),1));
d(acoth(x_),x_,n_):= d(atanh(x),x,n);

d((a_+x_)^n_,x_,n_):= if(n<0,n!*log(a+x));
d(sqrt(x_),x_, n_):=  (1/2)!/(1/2-n)!*x^(1/2-n);

d(1/(a_+log(x_)),x_,n_):=exp(-a)*li(1+n,exp(a)*x);
d(1/log(a_+x_),x_,n_):=if(isfree(a,x),li(1+n,a+x));
d(1/(x_^2+b_*x_+c_),x_,n_):=if(isfree(b,c,x),(-1)^n*b^n*n!*(x^2+b*x+c)^(-n-1));

d(sin(x_)/x_,x_,n_):=si(1+n,x);
d(cos(x_)/x_,x_,n_):=ci(1+n,x);
d(sin(a_*x_)/x_,x_,n_):=if(isfree(a,x),si(1+n,a*x));
d(cos(a_*x_)/x_,x_,n_):=if(isfree(a,x),ci(1+n,a*x));
d(sinh(a_*x_)/x_,x_,n_):=if(isfree(a,x),shi(1+n,a*x));
d(cosh(a_*x_)/x_,x_,n_):=if(isfree(a,x),chi(1+n,a*x));

d(x_,x_,n_) :=  If(n>1, 0, x^(1-n)/(1-n)!);
d(y_, x_=x0_, n_) := d(y,x,x0,n);
d(y_,x_,1) := d(y,x);
d(y_,x_,0) := y;


d(y_,x_=a_) := d(y,x,a,1);

d(integrate(y_,x_),x_) := y;
d(integrates(y_,x_),x_) := y;
d(integrates(y_,t_,a_,b_),x_) := block(f(t)=y,replace(f(t),t,b)*d(b,x)-replace(f(t),t,a)*d(a,x)+integrate(d(f(t),x),t,a,b));
d(integrate(y_,t_,a_,b_),x_) := replace(y,t,b)*d(b,x)-replace(y,t,a)*d(a,x)+integrate(d(y,x),t,a,b);
d(integrate(y(t_-x_),t_,a_,b_),x_) := y(b-x)*(d(b,x)-1)-y(a-x)*(d(a,x)-1);
d(integrate(y(x_-t_),t_,a_,b_),x_) := y(x-a)*(1-d(a,x))-y(x-b)*(1-d(b,x));

#d(infsums(y_,k_),x_,n_) := infsums(d(y,x,n),k);
d(infsums(x_^k_/k_!,k_),x_,n_) := infsums(x^k/k!,k);
d(infsums(x_^k_/k_!,k_),x_) := infsums(x^k/k!,k);
d(infsums(x_^(2k_)/(2k_)!,k_),x_) := infsums(x^(2k+1)/(2k+1)!,k);
d(infsums(1/(1+2k_)!*x_^(1+2k_),k_),x_) := infsums(x^(2k)/(2k)!,k);
d(infsums((-1)^k_/(2k_)!*x_^(2k_),k_),x_) := -infsums((-1)^k*x^(2k+1)/(2k+1)!,k);
d(infsums((-1)^k_/(1+2k_)!*x_^(1+2k_),k_),x_) := infsums((-1)^k*x^(2k)/(2k)!,k);

d(vector(x_,y_),xx_):=y/x;
d(mittag(a_,b_,x_), x_) := (mittag(a,b-1,x)-(b-1)*mittag(a,b,x))/(a*x);
d(mittag(a_,x_), x_) := mittag(a,a,x)/a;
d(Gamma(a_,m_,x_), x_) := Gamma(a+1,m,x);
d(Gamma(a_,x_,0),x_):=Gamma(a+1,x,0);
d(Gamma(n_, x_), x_) := -exp(-x)*x^(n-1);
d(zeta(x_), x_) := -sum(log(k)/k^x,k,2,oo);
d(zeta(n_,x_), x_) := -n*zeta(1+n,x);
d(zeta(n_,b_,x_), x_) := exp((1-b)*x)*x^(n-1)/(e^x-1)/Gamma(n);
d(eta(x_), x_) := -sum((-1)^k*log(k)/k^x,k,2,oo);
d(eta(n_,b_,x_), x_) :=  exp((1-b)*x)*x^(n-1)/(e^x+1)/Gamma(n);
d(eta(n_,x_), x_) :=  -n*eta(1+n,x);
d(En(n_,x_), x_) := -En(n-1,x);
#d(En(n_,x_), x_) := exp(-x)/(-x)^n;
#d(erf(n_,x_), x_) := n!/sqrt(pi)*exp(-x^n);
d(polylog(n_,x_), x_) := polylog(n-1,x)/x;
d(polylog(a_,n_,x_), x_) := -n*x^(a-1)/(e^x-n)/Gamma(a);
d(L(a_,b_,c_,x_), x_) := e^((1-c)*x)/(-a+e^(x))*x^(b-1)/Gamma(b);
d(when(a_,y_),x_):=when(a,d(y,x));
d(Beta(a_,b_,x_),x_):=x^(a-1)*(1-x)^(b-1);
d(Beta(x_,y_),x_):=Beta(x,y)*(psi(x)-psi(x+y));
d(Beta(x_,y_),y_):=Beta(x,y)*(psi(y)-psi(x+y));
d(beta(s_,x_),x_):=x^(s-1)/(e^(-x)+e^x)/Gamma(s);
d(harmonic(n_,1,x_),x_):=(1-x^n)/(1-x);
d(harmonic(n_,x_),x_):= -n*zeta(1+n,x+1);
d(Cl(a_,x_),x_):= (-1)^a*Cl(a-1,x);

d(W(x_),x_):=W(x)/(x+x*W(x));
d(Sophomore(x_),x_):=x^x;
d(Sophomore1(x_),x_):=1/x^x;
d(harmonic(x_),x_):= psi(1,x+1);
d(psi(x_),x_):= psi(1,x);
d(S(x_),x_):=sin(pi/2*x^2);
d(C(x_),x_):=cos(pi/2*x^2);
d(theta(y_),x_) := delta(x);
d(gauss(x_), x_) := -x*gauss(x);
d(Phi(x_),x_):=gauss(x);
d(inverseerf(x_), x_) := 1/2*sqrt(pi)*exp(inverseerf(x)^2);
d(erf(x_), x_) := 2/sqrt(pi)*exp(-x^2);
d(erfi(x_), x_) := 2/sqrt(pi)*exp(x^2);
d(Gamma(x_), x_) := Gamma(x)*psi(x);
d((x_)!, x_) := x!*psi(x+1);
d(loggamma(x_), x_) := psi(x);
d(li(x_),x_) := 1/log(x);
d(Ei(x_),x_) := exp(x)/x;
d(Ein(x_),x_) := (1-exp(-x))/x;

d(si(x_),x_) := sinc(x);
d(ci(x_),x_) := cos(x)/x;
d(tani(x_),x_) := tan(x)/x;
d(coti(x_),x_) := cot(x)/x;
d(csci(x_),x_) := csc(x)/x;
d(seci(x_),x_) := sec(x)/x;

d(asini(x_),x_) := asin(x)/x;
d(acosi(x_),x_) := acos(x)/x;
d(atani(x_),x_) := atan(x)/x;

d(shi(x_),x_) := sinh(x)/x;
d(chi(x_),x_) := cosh(x)/x;
d(tanhi(x_),x_) := tanh(x)/x;
d(cothi(x_),x_) := coth(x)/x;

d(sqrt(x_),x_):= 1/2*x^(-1/2);
d(cbrt(x_),x_):= 1/3*x^(-2/3);
d(log(x_),x_):=1/x;
d(log(abs(x_)),x_):=1/x;
d(abs(x_),x_):=1/sgn(x);
d(exp(x_),x_):=exp(x);

d(sin(x_),x_):= cos(x);
d(cos(x_),x_):= -sin(x);
d(tan(x_),x_):= sec(x)^2;
d(cot(x_),x_):= -csc(x)^2;
d(sec(x_),x_) := tan(x)*sec(x);
d(csc(x_),x_) := -cot(x)*csc(x);

d(asin(x_),x_) := (1-x^2)^(-1/2);
d(acos(x_),x_) := -(1-x^2)^(-1/2);
d(atan(x_),x_):= 1/(x^2+1);
d(atan2(x_,y_),y_) := -x/(x^2+y^2);
d(atan2(x_,y_),x_) := y/(x^2+y^2);
d(acot(x_),x_) := -1/(x^2+1);
d(asec(x_),x_) := 1/(x*sqrt(x^2-1));
d(acsc(x_),x_) := -1/(x*sqrt(x^2-1));

d(sinh(x_),x_):= cosh(x);
d(cosh(x_),x_):= sinh(x);
d(tanh(x_),x_) := sech(x)^2;
d(coth(x_),x_) := -csch(x)^2;
d(sech(x_),x_) := -tanh(x)*sech(x);
d(csch(x_),x_) := -coth(x)*csch(x);

d(asinh(x_),x_) := (x^2+1)^(-1/2);
d(acosh(x_),x_) := (x^2-1)^(-1/2);
d(atanh(x_),x_) := 1/(1-x^2);
d(acoth(x_),x_) := 1/(1-x^2);
d(asech(x_),x_) := -1/(x*sqrt(1-x^2));
d(acsch(x_),x_) := -1/(x*sqrt(1+x^2));

d(sgn(y_),x_) := 0;
d(csgn(y_),x_) := 0;
d(step(x_),x_):=0;
d(x_,x_):=1;

d(y_):=d(y,x);